Finding a C1 Function for Continuous f,g in Real Numbers

[johnson12]

Hello,I need some advice on a problem.

Let f,g:R\rightarrow R (where R denotes the real numbers) be two continuous functions, assume that f(x) < g(x) \forall x \neq 0 ,

and f(0) = g(0).Define A = \left\{(x,y)\neq (0,0): y&lt; f(x),x \in R\right\}<br />

B = \left\{(x,y)\neq (0,0): y&gt; g(x),x \in R\right\}<br />Let r,s:R^{2}\rightarrow R be C^{1},and show that there is a C^{1} function h defined on

R^{2}- \left\{(0,0)\right\} such that h(x,y) = r(x,y) on A and h(x,y) = s(x,y) on B.

It seems that this follows from partitions of unity, but I am not sure where to
start, any suggestions at all are helpful.
Thanks.

 

[johnson12]

Hello,I need some advice on a problem.

Let f,g:R\rightarrow R (where R denotes the real numbers) be two continuous functions, assume that f(x) &lt; g(x) \forall x \neq 0 ,

and f(0) = g(0).Define A = \left\{(x,y)\neq (0,0): y&lt; f(x),x \in R\right\}<br />

B = \left\{(x,y)\neq (0,0): y&gt; g(x),x \in R\right\}<br />


Let r,s:R^{2}\rightarrow R be C^{1},and show that there is a C^{1} function h defined on

R^{2}- \left\{(0,0)\right\} such that h(x,y) = r(x,y) on A and h(x,y) = s(x,y) on B.

It seems that this follows from partitions of unity, but I am not sure where to
start, any suggestions at all are helpful.
Thanks.

UPDATE: PROBLEM SOLVED